Using Prn1−1+rn−nt to Determine Regular Payment Amount
Use Prn1−1+rn−nt to determine the regular payment amount, rounded to the nearest dollar. Consider the following pair of mortgage loan options for a $150,000 mortgage. Which mortgage loan has the larger total cost (closing costs + the amount paid for points + total cost of interest)? By how much? Mortgage A: 15-year fixed at 9.25% with closing costs of $1500 and 1 point. Mortgage B: 15-year fixed at 7.5% with closing costs of $1500 and 5 points.
To determine the regular payment amount for each mortgage loan, we use the formula:
Prn / (1 – (1 + r)^(-n*t))
Where: P = principal amount (in this case, $150,000) r = monthly interest rate (annual rate divided by 12) n = number of payments (15 years = 180 months) t = number of points paid (for Mortgage A, t = 1; for Mortgage B, t = 5)
For Mortgage A: r = 9.25% / 12 = 0.00770833333 n = 180 t = 1
Plugging these values into the formula, we get:
150,0000.00770833333(1+0.00770833333)^180 / ((1+0.00770833333)^180 – 1) = $1,461.92
Rounding this to the nearest dollar, we get a regular payment amount of $1,462.
For Mortgage B: r = 7.5% / 12 = 0.00625 n = 180 t = 5
Plugging these values into the formula, we get:
150,0000.00625(1+0.00625)^180 / ((1+0.00625)^180 – 1) = $1,332.14
Rounding this to the nearest dollar, we get a regular payment amount of $1,332.
Now we can calculate the total cost for each mortgage loan, taking into account closing costs, points paid, and total interest paid over the life of the loan.
For Mortgage A: Closing costs = $1,500 Points paid = $150,000 * 1% = $1,500 Total interest paid = $1,461.92 * 180 – $150,000 = $90,546.60 Total cost = $1,500 + $1,500 + $90,546.60 = $93,546.60
For Mortgage B: Closing costs = $1,500 Points paid = $150,000 * 5% = $7,500 Total interest paid = $1,332.14 * 180 – $150,000 = $58,185.20 Total cost = $1,500 + $7,500 + $58,185.20 = $67,185.20
Therefore, Mortgage B has the larger total cost by $26,361.40 ($93,546.60 – $67,185.20).
